Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g<h$ there is $l\in G$ such that $lg=h$. For a totally ordered cancellative monoid the equalities $Z\left(RG\right)=Z\left(R\right)G$ and $\sigma \left(RG\right)=\sigma \left(R\right)G$ hold, $\sigma$ being Goldie’s torsion theory.

One of the results in my previous paper , preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \ge \sigma$, $\sigma$ being Goldie’s torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $𝔐=\{R,R/\sigma \left(R\right),R[x_1,\cdots ,x_n],R[x_1,\cdots ,x_n]/\sigma \left(R[x_1,\cdots ,x_n]\right),n<\omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover class if and only if this holds for an...

Recently, Rim and Teply , using the notion of $\tau$-exact modules, found a necessary condition for the existence of $\tau$-torsionfree covers with respect to a given hereditary torsion theory $\tau$ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau$-torsionfree and $\tau$-exact covers have been investigated in . The purpose of this note is to show that if $\sigma =({𝒯}_{\sigma },{ℱ}_{\sigma })$ is Goldie’s torsion theory and ${ℱ}_{\sigma }$ is a precover class, then ${ℱ}_{\tau }$ is a precover class...